A doom doo-wop band should be an impossibility: Doom is a plodding, earthbound style of metal while doo-wop harmonies are lighter than air. But against all odds, a combo called the Chemotherapy Marionettes pulls off that unlikely hybrid toward the end of “The Hollower.”

There are plenty of other counterintuitive juxtapositions in Liza Birkenmeier’s aggressively quirky new play at Access Theater, but they are not nearly as successful as that brief Chemotherapy Marionettes tune.

The main one brings together the laconic middle-aged lesbian Otto (Patrena Murray) and her hyperactive, voluble 16-year-old roommate, Bit (Reyna de Courcy), an odd couple thrown together by the arbitrary gods who rule Off Off Broadway.

How they ended up sharing a hovel ― You-Shin Chen’s set features the dirty dishes, empty soda bottles and crusty takeout containers that have come to represent American desperation onstage ― is not convincingly explained.

Admittedly, Ms. Birkenmeier makes clear from the start that she is not interested in conventional logic, or conventional storytelling, for that matter.

Otto, for instance, is pursued by the Pigman, a half-human, half-porcine podcaster (Ryan Wesley Stinnett) who drops by, usually uninvited, with Missy (ToniAnne DiFilippo), his helmet-wearing assistant. (The pig part is represented by a vaguely hoof-like glove.)

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Samuel Im (at right, with Ms. de Courcy) portrays a budding filmmaker known as the “Polanski of Claymation.”CreditHunter Canning

Bit totters on perilously high leopard-print heels, and sports a pink wig and mismatched clothes. She looks like a flamingo, walks like a chicken, and appears to live in a world governed by its own rules.

She plays in the aforementioned Marionettes, and enrolls a budding high school filmmaker named Wilkin Rush George III (Samuel Im) to help her make a video about Frenchwomen being shipped off to Canada in the 17th century.

Wilkin, a.k.a. “the Polanski of Claymation,” is relatively unfazed, though he does suggest that Bit is going to have to give her audience “something that allows them to like chill into your aesthetic.”

If only Ms. Birkenmeier herself had followed his advice. Instead, her play, directed by Kristy Dodson for New Light Theater Project, is a hermetically sealed experiment in extreme absurdism. It’s as if Wilkin was expressing the anticipated objections of critics wedded to oppressive Cartesian reason, and Ms. Birkenmeier chose to defy her own character and double down.

What’s most frustrating is that this young playwright, a finalist for the 2015 Relentless Award, can write strangely appealing segments ― not quite long enough to qualify as scenes, but fuller than mere exchanges. Some of her dialogue often pings with a darkly surreal humor, as when the Pigman tells Otto she can’t back out of their contract because it’s signed in blood. “Don’t worry, not your blood,” he says.

And then there is Otto herself, whom Ms. Murray endows with sad resignation. When she buys a box of industrial doughnuts for a late-night snack session, the play hits hard, without trying too hard.